This invention relates to a radar signal processor suitable for a vehicle-mounted array radar apparatus having two or more antenna components.
A conventionally known vehicle-mounted array radar apparatus judges presence or velocity of a preceding target which exists in a forward direction by estimating a power of a reflected wave from the target in order to prevent a collision and maintain the inter-vehicle distance with respect thereto (see a Japanese patent application publication number of which is 2004-198218).
A power estimating portion of such kind of the vehicle-mounted array radar apparatus may have two or more observation means, observation signal component extracting means, a sample correlation matrix computing portion, and a power estimating means according to maximum likelihood method.
A conventional array radar 1 as shown in FIG. 2 is comprised of observation means OBs 1 through K, observation signal component extracting means SDs 1 through K, sample correlation matrix computing means HG and power estimating means according to maximum likelihood method PG.
The observation means OB, p (array element p=1, 2, . . . K) obtains an observation signal S3 including information, such as an arrival azimuth of a reflected wave and a power thereof, from a transmitting signal which is a radar and a received signal received from an antenna. Observation signals XTp(t1), XTp(t2), XTp(t3), . . . , XTp(tM) which are sampled at times t1, t2, . . . tM are outputted from the observation means p (array element p=1, 2, . . . K), as shown in FIG. 2. M numbers of the observation signals which are sampled at times t1, t2, . . . tM is referred to as one snapshot. The observation means OBs, 1 through K correspond to array components 1 through K. A linear array has been known as a physical arrangement of the array components.
A case of a FMCW radar is now exemplarily mentioned. The observation means OB of a FMCW radar 4 is comprised of an oscillator 2, a transmitting amplifier 3, a transmitting antenna 5, a receiving antenna 6, a receiving amplifier 7, a distributor, a mixer 9, a filter 10, and A/D converter 11, as shown in FIG. 3. The oscillator 2 oscillates millimetric-wave signals as a transmitting signal S2 which are modulated such that the frequency increases and decreases linearly with passage of time to form a triangular wave form, and the millimetric-wave signals are emitted as the transmitted signal S2 via the transmitting amplifier 3 and the transmitting antenna 5. At the same time, the reflected wave of the transmitted signal S2 is received as a received signal S1 through the receiving antenna 6. After amplifying the signal S1 by the receiving amplifier 7, it is mixed with the transmitted signal S2 by the distributor and the mixer 9. The mixed signal is filtered by the filter 10, and converted into a digital signal by the A/D converter 11 so as to be sampled. This sampled signal is the observation signal S3 outputted by the observation means. Although the array radar has a plurality of the observation means OB as shown in FIG. 2, the transmitted signals S2 in the respective observation means are common.
A case of a pulse radar which is another instance is now mentioned. The observation means OB of a pulse radar 12 is comprised of an oscillator 13, the transmitting amplifier 3, the transmitting antenna 5, the receiving antenna 6, the receiving amplifier 7, a phase detector 15, the filter 10 and the A/D converter 11, as shown in FIG. 4. The oscillator 13 oscillates pulses which are obtained by dividing a signal having high frequency f0 every equal interval at a cycle of fr[Hz], and the transmitted signal S2 is emitted through the transmitting amplifier 3 and the transmitting antenna 5 (pulse transmission). At the same time, the reflected wave of the transmitted signal S2 is received as a received signal S1 through the receiving antenna 6. After amplifying the signal S1 by the receiving amplifier 7, it is detected by the phase detector 15, and is filtered by the filter 10. The signal is converted into the digital signal by the A/D converter 11 which is triggered by pulse transmission, and the digital signal is sampled. This sampling signal is the observation signal S3 which the observation means outputs. Similar to the FMCW radar, the transmitted signals S2 used in the respective observation means are common.
The observation signal component extracting means SD, p (array component p=1, 2, . . . K) as shown in FIG. 2 extracts an observation signal processing component XRp for subsequent stage from the observation signals S3, XTp(t1), XTp(t2), XTp(t3), . . . XTp(tM) of one snapshot which are outputted from the observation means OB, p.
For instance, a case of a FMCW radar is now mentioned. When a target having velocity V exists at a position of distance r, the observation signals S3, XTp(t1), XTp(t2), XTp(t3), . . . XTp(tM) of the FMCW radar include a frequency component of fB[Hz] as shown below.
[Expression 1]
                                          f            B                    =                                                                                          4                    ·                    Δ                                    ⁢                                                                          ⁢                  F                                                                      V                    C                                    ·                                      T                    m                                                              ·              r                        ±                                                            2                  ·                                      F                    0                                                                    V                  C                                            ·                              V                ⁢                                                                  [                Hz                ]                                                    ⁢                                  ⁢                  (                                                    +                                  :                                            ⁢                                                                                ⁢                                                                              ⁢              at              ⁢                                                          ⁢              the              ⁢                                                          ⁢              time              ⁢                                                          ⁢              of              ⁢                                                          ⁢              modulation              ⁢                                                          ⁢              by              ⁢                                                          ⁢              increasing              ⁢                                                          ⁢              frequency                        ,                                                  ⁢                                                  ⁢                                          -                                  :                                            ⁢                                                                                ⁢                                                                              ⁢              at              ⁢                                                          ⁢              the              ⁢                                                          ⁢              time              ⁢                                                          ⁢              of              ⁢                                                          ⁢              modulation              ⁢                                                          ⁢              by              ⁢                                                          ⁢              decreasing              ⁢                                                                                ⁢                                                                              ⁢              frequency                                )                                    (        1        )            where r denotes distance to a target, V denotes relative velocity of a target, Vc is light speed, Δ F is frequency deviation width of frequency modulation, Tm denotes cycle period of frequency modulation, and Fo denotes central transmitting frequency. (Correctly speaking, “distance r” is a half of a distance from the transmitting antenna 5 up to the receiving antenna 6 via a target. But, the distance r from the receiving antenna 6 is adopted as “distance r”, provided that the transmitting antenna 5 and the receiving antenna 6 are positioned at the same position. This explanation is applied to all descriptions in the present specification when referring to the distance up to a target r.) If relative velocity is neglected, the following relation which is shown hereinafter is given between distance r and frequency fB.[Expression 2]
                              f          B                =                                                            4                ·                Δ                            ⁢                                                          ⁢              F                                                      V                C                            ·                              T                m                                              ·                      r            ⁢                                                  [            Hz            ]                                              (        2        )            
If the frequency components fb corresponding to distance r are obtained from the observation signals XTp(t1), XTp(t2), XTp(t3), . . . XTp(tM) which are time series signals, the thus extracted are the observation signal components XRp for distance r. Fourier transformation or filtering with a band-pass filter is used as a method for extracting frequency components.
Besides, a case of a pulse radar is mentioned as another instance. If a target exists at the distance r, an echo from the target is observed in the observation signal S3 at a time
[Expression 3]
                              t          e                =                              2                          V              C                                ·          r                                    (        3        )            from a time when emitting a pulse where r denotes distance to the target, and Vc denotes light speed. If the observation signals S3, XTp(t1), XTp(t2), XTp(t3), . . . XTp(tM) are sampled at echo starting times te corresponding to distances r, the thus extracted are the observation signal components XRp for distance r. Such sampling may be conducted after simple sampling or average filtering.
The sample correlation matrix computing means computes a sample correlation matrix Cxx which is a sampled value of a correlation matrix Rxx which represents a correlation characteristics of the signals between the array components (coherence) from output signals XR1, XR2, . . . XRK of the observation signal component extracting means SD, 1 through K, which is obtained by each array component. An observation signal component vector XR is defined by next Expression.
[Expression 4]
                              X          R                ⁢                  =          Δ                ⁢                              [                                                                                                                                                                                                                                                                                                                                                              X                                                                              R                                        ⁢                                                                                                                                                                  ⁢                                        1                                                                                                                                                                                                                                                  ⋮                                                                                                                                                                                                                                                                          X                                Ri                                                                                                                                                                                                                        ⋮                                                                                                                                                              X                    RK                                                                        ]                    =                                    [                                                X                  R1                                ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  X                  Ri                                ⁢                                                                  ⁢                ⋯                ⁢                                                                  ⁢                                  X                  RK                                            ]                        T                                              (        4        )            
A superscript T represents a transposition. The correlation matrix Rxx is defined by next Expression. On this occasion, the correlation matrix Rxx is a complex matrix with K rows and K columns, and a component with i-th row and j-th column is represented by rxxqij.
[Expression 5]
                                                                        R                xx                            =                            ⁢                              [                                                                                                    r                                                  xx                          ⁢                                                                                                          ⁢                          1                          ⁢                          i                                                                                                            ⋯                                                                                      r                                                  xx                          ⁢                                                                                                          ⁢                          1                          ⁢                          j                                                                                                            ⋯                                                                                      r                                                  xx                          ⁢                                                                                                          ⁢                          1                          ⁢                          K                                                                                                                                                ⋮                                                                                                                                                                          ⋮                                                                                                                                                                          ⋮                                                                                                                          r                                                  xxi                          ⁢                                                                                                          ⁢                          1                                                                                                            ⋯                                                                                      r                        xxij                                                                                    ⋯                                                                                      r                        xxiK                                                                                                                                                ⋮                        ⁢                                                                                                                                                                                                                                                                          ⋮                                                                                                                                                                          ⋮                                                                                                                          r                                                  xxK                          ⁢                                                                                                          ⁢                          1                                                                                                            ⋯                                                                                      r                        xxKj                                                                                    ⋯                                                                                      r                        xxKK                                                                                            ]                                                                                                        =                Δ                            ⁢                            ⁢                              E                ⁡                                  [                                                            X                      R                                        ·                                          X                      R                      H                                                        ]                                                                                                                        =                Δ                            ⁢                            ⁢                              [                                                                                                    E                        ⁡                                                  [                                                                                    X                                                              R                                ⁢                                                                                                                                  ⁢                                1                                                                                      ·                                                          X                              Ri                              *                                                                                ]                                                                                                            ⋯                                                                                      E                        ⁡                                                  [                                                                                    X                                                              R                                ⁢                                                                                                                                  ⁢                                1                                                                                      ·                                                          X                              Rj                              *                                                                                ]                                                                                                            ⋯                                                                                      E                        ⁡                                                  [                                                                                    X                                                              R                                ⁢                                                                                                                                  ⁢                                1                                                                                      ·                                                          X                              RK                              *                                                                                ]                                                                                                                                                ⋮                                                                                                                                                                          ⋮                                                                                                                                                                          ⋮                                                                                                                          E                        ⁡                                                  [                                                                                                                    X                                                                  R                                  ⁢                                                                                                                                          ⁢                                  i                                                                                            ·                              XR                                                        ⁢                                                                                                                  ⁢                                                          1                              *                                                                                ]                                                                                                            ⋯                                                                                      E                        ⁡                                                  [                                                                                    X                                                              R                                ⁢                                                                                                                                  ⁢                                i                                                                                      ·                                                          X                              Rj                              *                                                                                ]                                                                                                            ⋯                                                                                      E                        ⁡                                                  [                                                                                    X                                                              R                                ⁢                                                                                                                                  ⁢                                i                                                                                      ·                                                          X                              RK                              *                                                                                ]                                                                                                                                                ⋮                                                                                                                                                                          ⋮                                                                                                                                                                          ⋮                                                                                                                          E                        ⁡                                                  [                                                                                    X                              RK                                                        ·                                                          X                                                              R                                ⁢                                                                                                                                  ⁢                                1                                                            *                                                                                ]                                                                                                            ⋯                                                                                      E                        ⁡                                                  [                                                                                    X                              RK                                                        ·                                                          X                              Rj                              *                                                                                ]                                                                                                            ⋯                                                                                      E                        ⁡                                                  [                                                                                    X                              RK                                                        ·                                                          X                              RK                              *                                                                                ]                                                                                                                    ]                                                                        (        5        )            
The superscript H represents a conjugate transpose. The superscript * represents a complex conjugate. E [ ] represents an operation for obtaining an expected value.
Since the sample correlation matrix Cxx is calculated in snapshots, the sample correlation matrix after a first snapshot is represented by Cxx(1), and the sample correlation matrix after a second snapshot is represented by Cxx(2), . . . and the sample correlation matrix after a m-th snapshot is represented by Cxx (m). Similar expression is used for the correlation matrix, the observation signal component, and the observation signal component vector etc., also.
With such kind of expression method, a correlation matrix observation value in the m-th snapshot Rtmp_xx(m) is calculated by next Expression.
[Expression 6]
                                                                                                              R                    tmp_xx                                    ⁡                                      (                    m                    )                                                  =                                ⁢                                  [                                                                                                                                          r                            tmp_xxl1                                                    ⁡                                                      (                            m                            )                                                                                                                      ⋯                                                                                                                          r                            tmp_xxlj                                                    ⁡                                                      (                            m                            )                                                                                                                      ⋯                                                                                                                          r                            tmp_xxlK                                                    ⁡                                                      (                            m                            )                                                                                                                                                              ⋮                                                                                                                                                                                          ⋮                                                                                                                                                                                          ⋮                                                                                                                                                                  r                            tmp_xxi1                                                    ⁡                                                      (                            m                            )                                                                                                                                                ⋯                          ⁢                                                                                                                                                                                                                                  r                            tmp_xxij                                                    ⁡                                                      (                            m                            )                                                                                                                      ⋯                                                                                                                          r                            tmp_xxiK                                                    ⁡                                                      (                            m                            )                                                                                                                                                              ⋮                                                                                                                                                                                          ⋮                                                                                                                                                                                          ⋮                                                                                                                                                                  r                            tmp_xxKi                                                    ⁡                                                      (                            m                            )                                                                                                                      ⋯                                                                                                                          r                            tmp_xxKj                                                    ⁡                                                      (                            m                            )                                                                                                                      ⋯                                                                                                                          r                            tmp_xxKK                                                    ⁡                                                      (                            m                            )                                                                                                                                ]                                            ⁢                                                                                                                                      =                Δ                            ⁢                            ⁢                                                                    X                    R                                    ⁡                                      (                    m                    )                                                  ·                                                                            X                      R                                        ⁡                                          (                      m                      )                                                        H                                                                                                        =                            ⁢                              [                                                                                                                                                          X                                                          R                              ⁢                                                                                                                          ⁢                              1                                                                                ⁡                                                      (                            m                            )                                                                          ·                                                                                                            X                                                              R                                ⁢                                                                                                                                  ⁢                                1                                                                                      ⁡                                                          (                              m                              )                                                                                *                                                                                                            ⋯                                                                                                                                            X                                                          R                              ⁢                                                                                                                          ⁢                              1                                                                                ⁡                                                      (                            m                            )                                                                          ·                                                                                                            X                                                              R                                ⁢                                                                                                                                  ⁢                                j                                                                                      ⁡                                                          (                              m                              )                                                                                *                                                                                                            ⋯                                                                                                                                            X                                                          R                              ⁢                                                                                                                          ⁢                              1                                                                                ⁡                                                      (                            m                            )                                                                          ·                                                                                                            X                              RK                                                        ⁡                                                          (                              m                              )                                                                                *                                                                                                                                                ⋮                                                                                                                                                                          ⋮                                                                                                                                                                          ⋮                                                                                                                                                                                X                                                          R                              ⁢                                                                                                                          ⁢                              i                                                                                ⁡                                                      (                            m                            )                                                                          ·                                                                                                            X                                                              R                                ⁢                                                                                                                                  ⁢                                1                                                                                      ⁡                                                          (                              m                              )                                                                                *                                                                                                            ⋯                                                                                                                                            X                                                          R                              ⁢                                                                                                                          ⁢                              i                                                                                ⁡                                                      (                            m                            )                                                                          ·                                                                                                            X                                                              R                                ⁢                                                                                                                                  ⁢                                j                                                                                      ⁡                                                          (                              m                              )                                                                                *                                                                                                            ⋯                                                                                                                                            X                                                          R                              ⁢                                                                                                                          ⁢                              i                                                                                ⁡                                                      (                            m                            )                                                                          ·                                                                                                            X                                                              R                                ⁢                                                                                                                                  ⁢                                K                                                                                      ⁡                                                          (                              m                              )                                                                                *                                                                                                                                                ⋮                                                                                                                                                                          ⋮                                                                                                                                                                          ⋮                                                                                                                                                                                X                            RK                                                    ⁡                                                      (                            m                            )                                                                          ·                                                                                                            X                                                              R                                ⁢                                                                                                                                  ⁢                                1                                                                                      ⁡                                                          (                              m                              )                                                                                *                                                                                                            ⋯                                                                                                                                            X                                                          R                              ⁢                                                                                                                          ⁢                              K                                                                                ⁡                                                      (                            m                            )                                                                          ·                                                                                                            X                                                              R                                ⁢                                                                                                                                  ⁢                                j                                                                                      ⁡                                                          (                              m                              )                                                                                *                                                                                                            ⋯                                                                                                                                            X                                                          R                              ⁢                                                                                                                          ⁢                              K                                                                                ⁡                                                      (                            m                            )                                                                          ·                                                                                                            X                                                              R                                ⁢                                                                                                                                  ⁢                                K                                                                                      ⁡                                                          (                              m                              )                                                                                *                                                                                                                    ]                                                                        (        6        )            
As a method of calculating the sample correlation matrix, a section average type and an exponential smoothing type are well-known.
In the method of the section average, an average value between correlation matrix observation values of snapshots which continues predetermined SSN [times] is used as the sample correlation matrix. A number of sampling SSN [times] for equalization relates to S/N improvement. If SSN becomes bigger, influence of noise in an observation signal is removed, so that S/N improves. One of methods of calculating is shown hereinafter.
[Expression 7]
                                          C            xx                    ⁡                      (            m            )                          =                              1            SSN                    ⁢                                    ∑                              j                =                0                                            SSN                -                1                                      ⁢                                          R                tmp_xx                            ⁡                              (                                  m                  -                  j                                )                                                                        (        7        )            
Although the above-mentioned refers to a case where the sample correlation matrix is renewed, synchronizing with the snapshot, the renewal cycle may be once SSN snapshot times. In such a case, the sample correlation matrix is renewed as shown hereinafter.
[Expression 8]
                                          C            xx                    ⁡                      (                          m              ′                        )                          =                              1            SSN                    ⁢                                    ∑                              j                =                1                            SSN                        ⁢                                          R                tmp_xx                            ⁡                              (                                                      SSN                    ·                                          (                                                                        m                          ′                                                -                        1                                            )                                                        +                  j                                )                                                                        (        8        )            where Cxx(m′) means the sample correlation matrix in the m′-th snapshot.
The method of the exponential smoothing is one for obtaining a renewed sample correlation matrix by respectively weighting the sample correlation matrix of the last snapshot and the correlation matrix observation value which is obtained in the present snapshot and adding both. A weight on the sample correlation matrix of the last snapshot is referred to as a forgetting factor, and is represented by α. At this time, a weight on the observation value of the correlation matrix which is obtained in the present snapshot is 1-α. A method of calculating with exponential smoothing is shown hereinafter.
[Expression 9]Cxx(m)=α·Cxx(m−1)+(1−α)·Rtmp—xx(m)  (9)
The number of sampling SSN [times] for equalization in the section average method which has been mentioned before and the forgetting factor α have the following relation as shown hereinafter in view of dispersion of the estimated value.
[Expression 10]
                    α        =                              SSN            -            1                                SSN            +            1                                              (        10        )            
Expression 10 is introduced by such a condition that dispersion of the element of the sample correlation matrix is equal in the section average method and the exponential smoothing method if each element of the observation value of the correlation matrix conform to a chi-square distribution of degree of freedom 2, but this is not detailedly mentioned. Then, SSN is made bigger as α approximates 1, so that the effect of the S/N improvement is made bigger. Therefore, the forgetting factor α is a parameter for adjusting the S/N improvement.
If Expression 9 is accepted as an IIR filter, a transient response performance on a change of the observation value of the correlation matrix is made better when α approximates zero (0), so that the forgetting factor α is a parameter for adjusting the transient response performance.
In the power estimating means according to the maximum likelihood method as shown in FIG. 2, the sample correlation matrix Cxx, an observation noise power PN, and reflected wave arrival azimuths θ 1, θ2, . . . , θD (number of reflected waves: D) are inputted, and the estimated values PS1, m1, . . . , PS2, m1  . . . PSD, m1 of the respective reflected wave power PS1, PS2, . . . , PSD are estimated according to the maximum likelihood method.
Concretely speaking, by determining an array response matrix V from the reflected wave arrival azimuths θ1, θ2, . . . , 74 D and calculating the next expression, a maximum likelihood estimated value Sf, m1 of the signal covariance matrix and the estimated values PS1, m1, PS2, m1, . . . , PSD, m1 of the respective reflected wave powers are obtained.
[Expression 11]Sf,ml=(VHV)−1VH[Cxx−PNI]V(VHV)−1 [PS1,ml. . . PSD,ml]=diag (Sf,ml)  (11)where diag means a process to extract a diagonal element, and H means a conjugate transpose.
In the power estimating means of the array radar apparatus 1 as shown in FIG. 2, observation noise power PN, number of reflected waves D, and the reflected wave arrival azimuths θ1, θ2, . . . , θD are explained as predetermined. But, some array radar apparatus may calculate observation noise power PN, number of reflected waves D, and reflected wave arrival azimuths θ1, θ2, . . . , θD from the sample correlation matrix Cxx. For instance, in a known method of deciding number of reflected waves D, this is determined from characteristic values of the sample correlation matrix with information standard, such as AIC or MDL, (see document: “OPTIMUM ARRAY PROCESSING Part IV of Detection, Estimation, and Modulation Theory”, Harry L. VanTrees, p. 830, 2002). And, in a known method of deciding the observation noise power PN, the characteristic values of the sample correlation matrix are arranged in a descending order, as λ1>λ2> . . . λD>λD+1 . . . >λ K and an average value between λD+1 and λK is determined as the observation noise power PN (see document: “OPTIMUM ARRAY PROCESSING Part IV of Detection, Estimation, and Modulation Theory”, Harry L. Van Trees, p. 1000, 2002). And, in a known method of deciding the reflected wave arrival azimuths θ1, θ2, . . . , θD, a pseudo angular power spectrum Pq (θ) is computed from the sample correlation matrix so as to be determined. The pseudo angular power spectrum shows an angular distribution of the pseudo power, and Pq (θ) shows the pseudo power which arrives from the azimuth θ, including an error. Known computing methods of the pseudo angular power spectrum are MUSIC, ESPRIT and the like (see document “Adaptive antenna technique” which has been published on Oct. 10, 2003 by Ohmsha written by Nobuo KIKUMA). The reflected wave is considered to arrive from an angle where the spectral intensity is intense in the pseudo angular power spectrum, so that a target is decided to be in this azimuth. In case of the pseudo angular power spectrum Pq(θ) as shown in FIG. 5, for instance, the arrival azimuth of the reflected wave is predicted to be at θ1, θ2 and θ3 [°] where spectrum strength is high.
Subsequently, it will now be briefly explained that output of Expression 11 becomes the estimated value by the maximum likelihood method, referring to document: “OPTIMUM ARRAY PROCESSING Part IV of Detection, Estimation, and Modulation Theory”, Harry L. Van Trees, p. 984˜, 2002.
The observation signal component vector XR is modeled by the array response matrix V, a signal component vector of reflected wave F, and observation noise vector N in the next expression.
[Expression 12]XR=V·F+N V[v(θ1) . . . v(θD)]F[s1, . . . ,sD]T  (12)
where v(θ) represents the array response vector and corresponds to the response at the time when emitting the reflected wave from arrival azimuth θ on the array radar. For instance, the array response vector of the linear array having K number of components which element interval is d is given by the next expression when a phase center is placed at the center of the array.
[Expression 13]
                              v          ⁡                      (            θ            )                          =                  [                                                                                                                                                                                                                                                                                                                                  exp                                    ⁡                                                                          (                                                                                                                        -                                          j                                                                                ⁢                                                                                                                              2                                            ⁢                                            π                                                                                    λ                                                                                ⁢                                                                                  d                                          ⁡                                                                                      (                                                                                                                                          K                                                -                                                1                                                                                                                                            2                                                ⁢                                                                                                                                                                                                                                                                                        )                                                                                                                          ⁢                                        sin                                        ⁢                                                                                                                                                                  ⁢                                        θ                                                                            )                                                                                                                                                                                                                                                                        exp                                    ⁡                                                                          (                                                                                                                        -                                          j                                                                                ⁢                                                                                                                              2                                            ⁢                                            π                                                                                    λ                                                                                ⁢                                                                                  d                                          ⁡                                                                                      (                                                                                                                                          K                                                -                                                3                                                                                                                                            2                                                ⁢                                                                                                                                                                                                                                                                                        )                                                                                                                          ⁢                                        sin                                        ⁢                                                                                                                                                                  ⁢                                        θ                                                                            )                                                                                                                                                                                                                                                                                                ⋮                                                                                                                                                                                                  exp                        ⁡                                                  (                                                      j                            ⁢                                                                                          2                                ⁢                                π                                                            λ                                                        ⁢                                                          d                              ⁡                                                              (                                                                                                      K                                    -                                    3                                                                                                        2                                    ⁢                                                                                                                                                                                                                )                                                                                      ⁢                            sin                            ⁢                                                                                                                  ⁢                            θ                                                    )                                                                                                                                                                                          exp                  ⁡                                      (                                          j                      ⁢                                                                        2                          ⁢                          π                                                λ                                            ⁢                                              d                        ⁡                                                  (                                                                                    K                              -                              1                                                                                      2                              ⁢                                                                                                                                                                            )                                                                    ⁢                      sin                      ⁢                                                                                          ⁢                      θ                                        )                                                                                ]                                    (        13        )            
λ is a wave length of reflected wave. The observation noise N and the observation noise power PN having the following relation.
[Expression 14]E[N·NH]=PNI  (14)
s1˜sD represent the respective signal components of the reflected waves of the arrival azimuths θ 1˜θD. A signal covariance matrix Sf is defined by the next expression.
[Expression 15]Sf=E[F·FH]  (15)
The signal covariance matrix Sf and the powers PS1, PS2, . . . , PSD of the respective reflected waves have the following relation. Therefore, in the estimation of power according to the maximum likelihood method, the object is to estimate the signal covariance matrix Sf.
[Expression 16]
                              S          f                =                                            [                                                          ⁢                                                                                          PS                      1                                                                            …                                                        …                                                                                        …                                                        ⋰                                                        …                                                                                        …                                                        …                                                                              PS                      D                                                                                  ⁢                                                          ]                        ⁢                                                  [                                          PS                1                            ⁢                                                          ⁢              …              ⁢                                                          ⁢                              PS                D                                      ]                    =                      diag            ⁡                          (                              S                f                            )                                                          (        16        )            
where diag( ) represents the process to extract the diagonal element from the matrix.
If the observation signal component vector XR is the random number of the complex Gaussian process, a conditional probability density distribution P(XR/Sf) of the observation signal component vector XR is given by the next expression.
[Expression 17]
                              P          ⁡                      (                                          X                R                            ❘                              S                f                                      )                          =                              1                          det              ⁡                              [                                  π                  ⁢                                                                          ⁢                                      R                    xx                                                  ]                                              ⁢          exp          ⁢                      {                                          -                                                      (                                                                  X                        R                                            -                                              E                        ⁡                                                  [                                                      X                            R                                                    ]                                                                                      )                                    H                                            ⁢                                                R                  xx                                      -                    1                                                  ⁡                                  (                                                            X                      R                                        -                                          E                      ⁡                                              [                                                  X                          R                                                ]                                                                              )                                                      }                                              (        17        )            
Therefore, a log likelihood function L(Sf) is given by the next expression.
[Expression 18]
                              L          ⁡                      (                          S              f                        )                          ⁢                  =          Δ                ⁢                              ln            ⁢                                                  ⁢                          P              ⁡                              (                                                      X                    R                                    ❘                                      S                    f                                                  )                                              =                      ln            ⁡                          (                                                1                                      det                    ⁡                                          [                                              π                        ⁢                                                                                                  ⁢                                                  R                          xx                                                                    ]                                                                      ⁢                exp                ⁢                                  {                                                            -                                                                        (                                                                                    X                              R                                                        -                                                          E                              ⁡                                                              [                                                                  X                                  R                                                                ]                                                                                                              )                                                H                                                              ⁢                                                                  R                        xx                                                  -                          1                                                                    ⁡                                              (                                                                              X                            R                                                    -                                                      E                            ⁡                                                          [                                                              X                                R                                                            ]                                                                                                      )                                                                              }                                            )                                                          (        18        )            
In the estimation of power according to the maximum likelihood method, the signal covariance matrix Sf which maximum is the likelihood function L(Sf) is obtained under the conditions of expressions 12 through 15 if observation noise power PN, number of reflected waves D, and arrival azimuths of reflected waves θ1, θ2, . . . , θD are given, and the powers PS1, PS2, . . . , PSD of the respective reflected waves are obtained from the Expression 15 with the obtained signal covariance matrix Sf as the maximum likelihood estimate Sf,m1. Supposing that E [XR] is not related to Sf and L (Sf) is transformed, the next expression is obtained.
[Expression 19]L(Sf)=−ln det[Rxx]−tr(Rxx−1Cxx)  (19)
where tr( ) represents the process to sum up the diagonal elements. In such a condition that L(Sf) is maximum,
[Expression 20]
                                                                        ∂                                  L                  ⁡                                      (                                          S                      f                                        )                                                                              ∂                                  S                  ij                                                      ⁢                          ❘                                                S                  f                                =                                  S                                      f                    ,                    ml                                                                                =          0                ,        i        ,                  j          =          1                ,        2        ,        …        ⁢                                  ,        D                            (        20        )            
where Sij is the component with i-th row and j-th column of the signal covariance matrix Sf. A partial differentiation by Sij of the first term of Expression 19 is given by the next expression.
[Expression 21]
                                                        ∂              ln                        ⁢                                                  ⁢                          det              ⁡                              [                                  R                  xx                                ]                                                          ∂                          S              ij                                      =                  tr          ⁡                      [                                                            v                  H                                ⁡                                  (                                      θ                    i                                    )                                            ⁢                              R                xx                                  -                  1                                            ⁢                              v                ⁡                                  (                                      θ                    j                                    )                                                      ]                                              (        21        )            
The second term is given by the next expression.
[Expression 22]
                                          ∂                          tr              ⁡                              [                                                      R                    xx                                          -                      1                                                        ⁢                                      C                    xx                                                  ]                                                          ∂                          S              ij                                      =                  tr          ⁡                      [                                          -                                                      v                    H                                    ⁡                                      (                                          θ                      i                                        )                                                              ⁢                              R                xx                                  -                  1                                            ⁢                              C                xx                            ⁢                              R                xx                            ⁢                              v                ⁡                                  (                                      θ                    j                                    )                                                      ]                                              (        22        )            
Therefore, the requirement of Expression 20 is
[Expression 23]vH(θi)[Rxx−1CxxRxx−Rxx−1]v(θj)=0, i,j=1,2, . . . D  (23)
or
[Expression 24]VH[Rxx−1CxxRxx−Rxx−1]V=0  (24)
From the following Expression 25 obtained from Expressions 12, 14, and 15,
[Expression 25]Rxx=VSfVH+PNI  (25)
the following expression is obtained.
[Expression 26]
                              R          xx                      -            1                          =                              1                          P              N                                ⁡                      [                          I              -                                                                    V                    ⁡                                          [                                                                                                    S                            f                                                    ⁢                                                      V                            H                                                    ⁢                          V                                                +                                                                              P                            N                                                    ⁢                          I                                                                    ]                                                                            -                    1                                                  ⁢                                  S                  f                                ⁢                                  V                  H                                                      ]                                              (        26        )            
Therefore, when transforming Expression 24 with Expression 26, the next expression is obtained.
[Expression 27][SfVHV+PNI]−1VH[Cxx−Rxx]V[SfVHV+PNI]−1=0  (27)
As a necessary and sufficient condition for formation of Expression 27, the next expression is obtained.
[Expression 28]VH[Cxx−Rxx]V=0  (28)
Substituting Expression 25 for Expression 28, the following expression is obtained.
[Expression 29](VHV)Sf(VHV)=VH[Cxx−PnI]V  (29)
Since the array response matrix V, the sample correlation matrix Cxx, and the observation noise power PN are already known in Expression 29, Sf the maximum of which is the likelihood function is a value which satisfys the linear matrix equation in Expression 28. When showing Expression 29 as the matrix element again,
[Expression 30]
                                                                                          p                  ⁢                                      -                                    ⁢                  th                                                                                    row                                              ⁢                                    [                                                          ⁢                                                                                                                                                                                                                                                                        ⋮                                                                                                                                                          …                                                                                                                                                                                                                                                                                            ⋮                                                                                                                                                                                                                                                                                            …                                                        …                                                                                                      ∑                                                  i                          =                                                      1                            ⁢                                                                                                                  ⁢                            …                            ⁢                                                                                                                  ⁢                            D                                                                                              ⁢                                                                        ∑                                                      j                            =                                                          1                              ⁢                                                                                                                          ⁢                              …                              ⁢                                                                                                                          ⁢                              D                                                                                                      ⁢                                                                                                            v                              ⁡                                                              (                                                                  θ                                  p                                                                )                                                                                      H                                                    ⁢                                                      v                            ⁡                                                          (                                                              θ                                i                                                            )                                                                                ⁢                                                      S                            ij                                                    ⁢                                                                                    v                              ⁡                                                              (                                                                  θ                                  j                                                                )                                                                                      H                                                    ⁢                                                      v                            ⁡                                                          (                                                              θ                                q                                                            )                                                                                                                                                                                …                                                        …                                                                                                                                                                                                                                                                                            ⋮                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ⋮                                                                                                                                                                                                                                                                  ⁢                                                          ]                                      q              ⁢                              -                            ⁢              th              ⁢                                                          ⁢              column                                      =                                  ⁢                                  ⁢                                  ⁢                  [                                          ⁢                                                                                                                                                                                                                ⋮                                                                                                                          …                                                                                                                                                                                                                                ⋮                                                                                                                                                                                                                                …                                            …                                                                                                                                v                        ⁡                                                  (                                                      θ                            p                                                    )                                                                    H                                        ⁡                                          [                                                                        C                          xx                                                -                                                                              P                            N                                                    ⁢                          I                                                                    ]                                                        ⁢                                      v                    ⁡                                          (                                              θ                        q                                            )                                                                                                  …                                            …                                                                                                                                                                                                                                ⋮                                                                                                                                                                                                                                                                                                                                                                                            ⋮                                                                                                                                                                                                          ⁢                                          ]                                    (        30        )            Sij is the element of i-th row and j-th column of Sf
In other words, in the power estimation according to the maximum likelihood method, the estimated value is obtained by solving the simultaneous equations in the number of square of D which is combination of the respective elements in p rows and q columns (p=1 . . . D, q=1 . . . D) as shown in Expression 30 for variables Sij i=1 . . . D, j=1 . . . D. In Expression 11, an inverse matrix of VHV is calculated from the array response matrix to be determined in Expression 12, and Sf satisfying Expression 29, that is, the maximum likelihood estimate Sf,m1, is obtained.
The above-mentioned is the structure of the power estimating portion of the vehicle-mounted array radar apparatus. In the array radar apparatus for collision avoidance and traveling maintaining inter-vehicle distance, real-time processing at high speed is required by its nature.
In estimation of power according to the maximum likelihood method with Expression 11, but, it may be difficult to embody because of the operation of inverse matrix with large computation. The computation volume of the inverse matrix operation according to the Gaussian elimination method is indicated in Table 1. When thinking multiplication having a number of operation times and large amount of computer resources as a standard, computing amount, a cube of matrix order is necessary for the operation of inverse matrix.
[Expression 11]Sf,ml=(VHV)−1VH[Cxx−PNI]V(VHV)−1 [PS1,ml . . . PSD,ml]=diag(Sf,ml)  (11)
TABLE 1Computing Volume for Inverse Matrix of Matrix havingn orders (Case of Gaussian Elimination Method)Computing volumeOperator(times)OrderAddition and4/3 · n3 − 3/2 · n2 + 1/6 · n0 (n3)SubtractionMultiplication4/3 · n3 − 3/2 · n2 + 1/6 · n0 (n3)Division3/2 · n2 − 1/2 · n0 (n2)
The invention provides the radar signal processor having a power estimating means for reducing computing volume by introducing the approximating expression which is obtained by simplifying Expression 11, for real-time processing.